De Gea became a free agent at the end of June and was locked in negotiations to extend his stay at Old Trafford but no agreement has been reached.
The Spain international is set to make way for Andre Onana, who looks likely to join from Inter Milan in the coming days.
De Gea joined Manchester United from Atletico Madrid in the summer of 2011 and made 545 appearances, winning one Premier League title, the Europa League, two League Cups and one FA Cup.
The 32-year-old kept the most clean sheets in the Premier League last season with 17.
De Gea won Manchester United's player of the year four times during his stay at Old Trafford.
"I just wanted to send this farewell message to all Manchester United supporters," De Gea wrote on social media.
"I would like to express my unwavering gratitude and appreciation for the love from the last 12 years.
"We've achieved a lot since my dear Sir Alex Ferguson brought me to this club. I took incredible pride every time I pulled on this shirt, to lead the team, to represent this institution, the biggest club in the world was an honour only bestowed upon a few lucky footballers."
"It's been an unforgettable and successful period since I came here," De Gea said.
"I didn't think from leaving Madrid as a young boy we would achieve what we did together. Now, it's the right time to undertake a new challenge, to push myself again in new surroundings.
"Manchester will always be in my heart, Manchester has shaped me and will never leave me. We've seen it all."
Manchester United manager Erik Ten Hag added: Ten Hag said: "It takes great quality and character to reach the level of even playing one game for Manchester United. To do it 545 times over 12 years is a special achievement, particularly in the goalkeeping position where every game puts you in the spotlight.
"To have won Player of the Year awards from both the fans and his team-mates, each on four occasions, shows the level of his performance and he will always be remembered as one of the very best goalkeepers in the history of the club.
"I am personally grateful for the 25 clean sheets he helped provide last season and his overall contribution on the pitch during my first year at the club. All the players and staff send him our very best wishes for the next phase of his outstanding career."
